Bayes and Kalman Filter Day 22 Bayes and Kalman Filter 2/23/2019

Combining Two Noisy Measurements recall from the last lecture that the minimum variance estimate for combining two noisy measurements claim: the estimate is a special case of the discrete Kalman filter algorithm previous measurement measurement difference Kalman gain 2/23/2019

Discrete Kalman Filter estimates the state x of a discrete-time controlled process that is governed by the linear stochastic difference equation with a measurement plant model process model measurement model observation model 2/23/2019

Components of a Kalman Filter Matrix (nxn) that describes how the state evolves from t to t-1 without controls or noise. Matrix (nxl) that describes how the control ut changes the state from t to t-1. Matrix (kxn) that describes how to map the state xt to an observation zt. Random variables representing the process and measurement noise that are assumed to be independent and normally distributed with covariance Rt and Qt respectively.

Kalman Filter Algorithm Algorithm Kalman_filter( mt-1, St-1, ut, zt): Prediction: Correction: Return mt, St

Combining Two Noisy Measurements combining two noisy measurements of a fixed scalar quantity is a static 1D-state estimation problem the state does not evolve as a function of time and does not depend on any control input our measurements are direct (noisy) measurements of the state 2/23/2019

Combining Two Noisy Measurements start by initializing the Kalman filter with the first measurement and its variance now substitute into the Kalman filter algorithm estimated state estimated state covariance 2/23/2019

Plant or Process Model describes how the system state changes as a function of time, control input, and noise state at time t control inputs at time t process noise at time t (assumed Gaussian with covariance Rt) state transition model or matrix at time t control-input model or matrix at time t note that the model is linear and assumes additive Gaussian noise 2/23/2019

Example: Omnidirectional Robot an omnidirectional robot is a robot that can move in any direction (constrained in the ground plane) http://www.youtube.com/watch?v=DPz-ullMOqc http://www.engadget.com/2011/07/09/curtis-boirums-robotic-car- makes-omnidirectional-dreams-come-tr/ if we are not interested in the orientation of the robot then its state is simply its location 2/23/2019

Example: Omnidirectional Robot a possible choice of motion control is simply a change in the location of the robot with noisy control inputs 2/23/2019

Measurement Model describes how sensor measurements vary as a function of the system state sensor measurement at time t sensor noise at time t (assumed Gaussian with covariance Qt) observation model or matrix notice that the model is linear and assumes additive Gaussian noise 2/23/2019

Kalman Filter the Kalman filter is a provably optimal (in terms of least- squared error) algorithm for fusing sensor measurements to produce an estimate of the state and the state covariance state at time t state covariance at time t 2/23/2019

Kalman Filter the Kalman filter estimates a process in two stages prediction: current state and state covariance estimates are projected forward in time to predict the new state and state covariance “time update equations” correction: the sensor measurements are incorporated into the predicted state to obtain improved estimates of the state and state covariance “measurement update equations” time update (predict) measurement update (correct) 2/23/2019

Kalman Filter Algorithm Initialization choose (guess) initial values for mean state and state covariance estimates 2/23/2019

Kalman Filter Algorithm Prediction: predict the next state using the plant model predicted state covariance grows (because we are not incorporating the sensor measurements yet) covariance of the plant noise 2/23/2019

Kalman Filter Algorithm Correction: correct the predicted state using the sensor measurement expected value of measurements (from measurement model) difference between actual and expected measurements measurement covariance Kalman gain 2/23/2019

Kalman Filter Algorithm State and state covariance: new state estimate incorporating most recent measurement new state covariance estimate 2/23/2019